cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A271469 G.f. satisfies A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5).

Original entry on oeis.org

1, 1, 4, 23, 155, 1142, 8910, 72350, 605056, 5175866, 45077560, 398348733, 3562916317, 32192775763, 293410452560, 2694283228653, 24902681767987, 231496130358758, 2162985033344112, 20301976721356134, 191336242071696514, 1809916398759630481, 17178063381786563194, 163536967014934201972, 1561247114394683682834, 14943175106109268856975
Offset: 0

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Author

Paul D. Hanna, Apr 08 2016

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 155*x^4 + 1142*x^5 + 8910*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 54*x^3 + 372*x^4 + 2778*x^5 + 21873*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 663*x^4 + 5025*x^5 + 39970*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 144*x^3 + 1041*x^4 + 8016*x^5 + 64470*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 205*x^3 + 1520*x^4 + 11901*x^5 + 96850*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 278*x^3 + 2115*x^4 + 16848*x^5 + 138816*x^6 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 364*x^3 + 2842*x^4 + 23044*x^5 + 192325*x^6 +...
where A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5),
and A(x)^2 = 1 + x*(A(x)^3 + A(x)^6),
and A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7),
and A(x)^4 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^8),
and A(x)^5 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^9), etc.
The g.f. satisfies A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3) where
F(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + 51*x^7 +...+ A001006(n-1)*x^n +...
is a g.f. of the Motzkin numbers (A001006, shifted right 1 place).
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 366*x^5 + 2148*x^6 +...+ A219537(n)*x^n +...
satisfies G(x) = 1 + x*(G(x)^2 - G(x)^3 + G(x)^4).

Links

Crossrefs

Cf. A363573, A001006, A370474.
Cf. A001006, A106228, A219537, A364765, A365194.
Cf. A001764, A370476.

Programs

  • Mathematica
    CoefficientList[(1/x*InverseSeries[Series[8*x^4/(1 + x - Sqrt[1 - 2*x - 3*x^2])^3, {x, 0, 20}], x])^(1/3), x] (* Vaclav Kotesovec, Apr 16 2016 *)
  • PARI
    /* Formula A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5): */
{a(n)=local(A=1); for(i=1, n, A=1+x*(A^3-A^4+A^5) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Formula using Series Reversion involving Motzkin numbers: */
{a(n)=local(A=1); A=(1+x-sqrt(1-2*x-3*x^2+x^3*O(x^n)))/(2*x); polcoeff( (1/x*serreverse(x/A^3))^(1/3), n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x*(A(x)^3 + A(x)^6).
(2) A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7).
Let F(x) = (1+x - sqrt(1 - 2*x - 3*x^2)) / (2*x), then g.f. A(x) satisfies:
(3) A(x) = ( (1/x)*Series_Reversion(x/F(x)^3) )^(1/3),
(4) A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3),
where F(x) = 1 + x*M(x) such that M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
Let G(x) = 1 + x*(G(x)^2 - G(x)^3 + G(x)^4), then g.f. A(x) satisfies:
(5) A(x) = (1/x)*Series_Reversion(x/G(x)),
(6) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
where G(x) is the g.f. of A219537.
a(n) ~ sqrt((34 + (34102 - 8262*sqrt(17))^(1/3) + (34102 + 8262*sqrt(17))^(1/3)) / 1632) * ((28 + (513243 - 4131*sqrt(17))^(1/3)/3 + (19009 + 153*sqrt(17))^(1/3)) / 8)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Apr 16 2016
D-finite recurrence: 8*n*(2*n-1)*(4*n-1)*(4*n+1)*(204*n^4 - 1341*n^3 + 3191*n^2 - 3286*n + 1242)*a(n) = 12*(45696*n^8 - 391776*n^7 + 1376164*n^6 - 2580579*n^5 + 2808064*n^4 - 1797694*n^3 + 651566*n^2 - 119476*n + 8160)*a(n-1) - 6*(n-2)*(29376*n^7 - 237168*n^6 + 760044*n^5 - 1236774*n^4 + 1082233*n^3 - 496791*n^2 + 108530*n - 8400)*a(n-2) + 9*(n-3)*(n-2)*(3*n-8)*(3*n-4)*(204*n^4 - 525*n^3 + 392*n^2 - 111*n + 10)*a(n-3). - Vaclav Kotesovec, Apr 16 2016
From Seiichi Manyama, Aug 06 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(3*n+k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(5*n-2*k,n-1-k) for n > 0. (End)
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A370474. - Seiichi Manyama, Mar 31 2024
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n/2+3*k/2+1/2,n)/(3*n+3*k+1). - Seiichi Manyama, Apr 04 2024

A259757 G.f. A(x) satisfies A(x)^2 = 1 +x + x*A(x)^5.

Original entry on oeis.org

1, 1, 2, 8, 35, 169, 862, 4575, 24999, 139700, 794684, 4586377, 26788423, 158054285, 940603900, 5639481930, 34032324940, 206550445064, 1259975808104, 7720835953740, 47504293931640, 293357473042545, 1817649401577760, 11296505623845080, 70402438290940450, 439888817329463279, 2755010697928837222, 17292270772076728414
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

Terms appear to equal A011791, apart from offset and an initial 1.
Note that the function G(x) = 1 + x*G(x)^2 (g.f. of A000108) also satisfies this condition: G(x) = 1/G(-x*G(x)^3).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 35*x^4 + 169*x^5 + 862*x^6 + 4575*x^7 + 24999*x^8 + 139700*x^9 + 794684*x^10 +...
where A(x)^2 = 1+x + x*A(x)^5 and
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 90*x^4 + 440*x^5 + 2266*x^6 + 12110*x^7 + 66525*x^8 + 373320*x^9 + 2130865*x^10 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 90*x^3 + 440*x^4 + 2266*x^5 + 12110*x^6 + 66525*x^7 + 373320*x^8 + 2130865*x^9 + 12332512*x^10 +...
OTHER RELATIONS.
Let B(x) be defined by B(x*A(x)) = x, then
B(x) =  x - x^2 - 3*x^4 - 3*x^5 - 22*x^6 - 50*x^7 - 240*x^8 - 763*x^9 - 3234*x^10 - 11880*x^11 - 48831*x^12 +...
Let C(x) be defined by C(x*A(x)^2) = A(x), then
C(x) = 1 + x + 3*x^3 - 3*x^4 + 22*x^5 - 50*x^6 + 240*x^7 - 763*x^8 + 3234*x^9 - 11880*x^10 + 48831*x^11 +...

Links

Crossrefs

Cf. A011791, A295537, A295538, A366452.
Cf. A370472, A370473, A370476.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A = sqrt(1+x + x*A^5 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies [from Paul D. Hanna, Nov 27 2017]:
(1) 1 + Series_Reversion( x/(1 + 2*x + 4*x^2 + 3*x^3 + x^4) ).
(2) F(A(x)) = x such that F(x) = -(1 - x^2)/(1 + x^5).
(3) A(x) = 1 / A(-x*A(x)^3).
Recurrence: 3*(n-2)*(n-1)*n*(3*n - 1)*(3*n + 1)*a(n) = 6*(n-2)*(n-1)*(2*n - 1)*(3*n - 2)*(3*n - 1)*a(n-1) + 10*(n-2)*(41*n^4 - 164*n^3 + 200*n^2 - 72*n + 3)*a(n-2) + 100*(n-3)*n*(2*n - 3)*(2*n^2 - 6*n + 3)*a(n-3) + 125*(n-4)*(n-3)*(n-1)^2*n*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ 3^(n - 5/2) * 5^n * sqrt((15 + 4*10^(1/3) + 2*10^(2/3))/Pi) / (2*n^(3/2) * (10^(2/3) + 4*10^(1/3) - 11)^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017
D-finite with recurrence 9*n*(3*n-1)*(3*n+1)*a(n) -6*(3*n-2) *(48*n^2-115*n+83)*a(n-1) +15*(n-1) *(17*n^2-169*n+254)*a(n-2) +50 *(n-3)*(194*n^2-971*n+1200) *a(n-3) +125*(n-4) *(143*n^2-856*n+1265) *a(n-4) +2500*(n-5) *(5*n^2-35*n+59)*a(n-5) +3125*(n-5)*(n-6)*(n-3)*a(n-6)=0. - R. J. Mathar, Nov 16 2023
From Seiichi Manyama, Apr 04 2024: (Start)
G.f. A(x) satisfies A(x) = 1 + x * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k/2+1/2,n)/(5*k+1). (End)

A370472 G.f. satisfies A(x) = 1 + x * A(x) * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).

Original entry on oeis.org

1, 1, 3, 15, 88, 565, 3844, 27228, 198670, 1482981, 11271117, 86926262, 678568982, 5351340410, 42570335161, 341201704970, 2752693408051, 22335989938093, 182166978172055, 1492496248447713, 12278191839580716, 101382009468089580, 839932374157895727
Offset: 0

Views

Author

Seiichi Manyama, Mar 31 2024

Keywords

Crossrefs

Cf. A106228, A370471.
Cf. A370473, A370476.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial(n/2+5*k/2+1/2, n)/(n+5*k+1));

Formula

G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x * A(x) * (1 + A(x)^5).
(2) A(x) = sqrt(B(x)) where B(x) is the g.f. of A370471.
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n/2+5*k/2+1/2,n)/(n+5*k+1).

A370473 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).

Original entry on oeis.org

1, 1, 4, 25, 185, 1501, 12914, 115723, 1068505, 10094770, 97117624, 948181724, 9370734322, 93562986440, 942385174150, 9563720899515, 97696642766654, 1003789888620166, 10366477185870960, 107548800153957745, 1120374840689934195, 11714707429579539268
Offset: 0

Views

Author

Seiichi Manyama, Mar 31 2024

Keywords

Crossrefs

Cf. A000108, A219537.
Cf. A370472, A370476.
Cf. A366401.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial(n+5*k/2+1/2, n)/(2*n+5*k+1));

Formula

G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x * A(x)^2 * (1 + A(x)^5).
(2) A(x) = sqrt(B(x)) where B(x) is the g.f. of A366401.
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+5*k/2+1/2,n)/(2*n+5*k+1).
Showing 1-4 of 4 results.

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